Introduction
Recently, all of us were witnesses of the terrible tsunami that occurred on
11 March 2011 off the coast of Japan. The tsunami in Japan resembled the
disaster in the Indian Ocean in 2004. These mega-events prompted serious
efforts to address the mitigating strategies against the threats posed by tsunamis.
An increasing reliability of tsunami prediction can partially be achieved
by means of numerical modeling, which makes it possible to estimate the expected propagation, as
well as the run up, wave heights and arrival times of a tsunami on the coast that
could be subject to risk. An important part of tsunami simulation is to gain
some insight into the tsunami source. The modern tsunami early warning
systems conventionally employ seismic methods to determine the source
parameters. The approaches based on inversion of remote measurements of
sea-level data have some advantages because seismic data are not available shortly after an event and are often imprecisely translated to tsunami
data. Furthermore, tsunami wave propagation can be simulated more precisely
than that of seismic waves.
Among the mathematical approaches based on inversion of near-field
water-level data are the methods based on Green's functions technique (GFT)
, least square inversion combined with the GFT
and an optimization approach . A priori
information from seismic data about a tsunami source played an essential role
in the inversion method of Satake. One of the main advantages of the second
group of methodologies is that they do not require a priori assumption of a
fault plane solution. The first and second approaches are based on the
linear shallow water theory, but the third one allows us to use the nonlinear
shallow water equations or other appropriate equation sets. These methods
have been widely applied in further studies of tsunami problems with various
modifications .
Among the recent studies dealing with the inverse tsunami source problem, it
is necessary to cite an approach which is successfully applied now and which
is a method of direct sorting combined with the minimal residual method
. In brief, this method determines the unknown
coefficients of a set of the typical unit tsunamigenic sub-faults disposed at
the subduction zone so that residuals between the calculating data from such
combination of unit sub-faults and the real DARTR data will be as small as
possible in a least squares sense.
The developed numerical inversion technique based on least square inversion
and truncated SVD (singular value decomposition) approach is here described
to reconstruct the initial tsunami waveform (tsunami source)
in a tsunami source area based on inversion of remote water-level
measurements (marigrams). This inversion method was first described in its
fundamentals in the Russian scientific journals . Theoretical
considerations of such a methodology for a linear long-wave model was
discussed by .
A direct problem of tsunami wave propagation is considered within the scope
of the linear shallow-water theory. Numerical simulation is based on a finite
difference algorithm on staggered grids. This ill-posed inverse problem of
recovering initial tsunami waveforms is regularized by means of least square
inversion using a truncated SVD approach. As a result of the numerical
process an r-solution is obtained . The proposed
method allows one to control the numerical instability of the solution and to
obtain an acceptable result in spite of the so-called ill-posedness of the
problem. The efficiency of the inversion is defined by the relative errors of
tsunami source reconstruction. Analysis of the singular spectrum of a matrix
obtained allows one to predict the efficiency of inversion by using records
produced at a given set of receivers.
One of the substantial advantages of this method is that it is completely
independent of any particular source model and its distinguishing feature is
the possibility to estimate the capability of a certain observation system to
recover the tsunami source.
Based on the properties of the inverse operator studied numerically, we tried to
answer the following questions: (1) what minimum number of marigrams should
be used to reconstruct a tsunami source well enough; (2) where should the
recorders (receivers) of the water-level oscillations be disposed
relative to potential source area; (3) how accurately can a tsunami source be
reconstructed based on recordings with a given monitoring system; (4) is it
possible to improve the quality of reconstructing a tsunami source by
distinguishing the most informative part of the observation system?
In order to answer these questions within the approach proposed, we have
carried out a series of numerical experiments with synthetic data and
different computational domains. The results of the numerical simulations
have shown a promising outlook of this approach.
Models
Mathematically, the problem of recovering the initial tsunami waveform in a
source area is formulated as the determination of the spatial distribution of
an oscillation source using remote measurements on a finite set of points
(hereafter called receivers). Let us consider a coordinate system
xyz and direct the axis z downwards. The plane {z = 0} corresponds to
an undisturbed water surface. The curvature of the Earth is neglected. Let
Π = {(x; y): 0 ≤ x ≤ X; 0 ≤ y ≤ Y} is a rectangular domain on
the plane {z = 0}. We denote as Φ the aquatic part of Π with
arbitrary solid boundaries Γ and straight-line sea boundaries.
The model bottom topography having some basic morphological features
typical of the island arc regions.
One of the examples of spatial layout of this statement with straight-line
coastal boundaries is represented in Fig. and will be
considered in Sect. 5.2. In this case, the domain Φ coincides with
the domain Π. The interaction between the wave and the coast is not
considered in this study. Our numerical model is based on the shallow water
theory. On addition, we look for a solution only in a constrained region. The
source area is assumed to be known from seismological data.
Let Ω = {(x, y) : x1 ≤ x ≤ xM; y1 ≤ y ≤ yN} be a tsunami source,
Ω ⊂ Φ ⊆ Π. Let η(x, y, t) be the function of the
water surface elevation relative to the mean sea level. This function is
considered to be a solution of the linear shallow water equation:
ηtt=div(gh(x,y)gradη)
with the initial conditions:
η|t=0=φ(x,y);ηt|t=0=0
with the completely reflecting conditions on the continental coasts:
∂η∂n|Γ=0.
Fully absorbing boundary conditions (ABC) of second order
accuracy are implied on the open boundaries. The acceleration of gravity is
denoted as g and the wave phase velocity is defined as
c(x, y) = gh(x,y). The tsunami wave is assumed to be triggered by a
sudden vertical displacement φ(x, y) of the sea floor in the target
domain Ω. The variable h(x, y) is the water depth relative to the
mean sea level.
The set-up inversion experiments are substantially different in the function
h(x, y) which varies from h(x, y) = const to a special h(x, y) = h(x) modeling
the shelf zone and, finally, to the real bathymetry of the Peru subduction zone.
The observational data are water level records which are assumed to be known
at a set of points M = {(xi, yi), i = 1, …, P} in the domain Φ:
ηxi,yi,t=η0xi,yi,t,xi,yi∈M.
One can also assume that the set of points M belongs to some line γ
without self-crossing in the domain Φ that is necessary only for
theoretical purposes.
Inversion method
In short, this method is as follows. Let us denote by A the linear
operator of the Cauchy problem presented by Eqs. ()–()
and trace its solution on the line γ(s). Under an appropriate
assumption on the functions h(x, y), φ(x, y) and the line
γ(s) (this assumption does not necessarily hold in the experiments),
by means of the standard technique of embedding theorems it was proved
that the operator A : L2(Ω) → L2(γ(s) × (0, T)) is compact and, therefore, does
not possess a bounded inverse. Thus, Eqs. ()–() are now
reduced to the following equation:
Aφ=η0(s,t),
where
η0(s,t)=η(x(s),y(s),t),(x(s),y(s))∈γ(s),0≤t≤T,0≤s≤L.
As was shown by , the above inverse problem has a unique
solution only if the source function allows factorization in the temporal and
spatial variables.
The inverse problem in question can now be formulated as the problem of
solving a linear operator equation of the first kind. Its solution will be
sought for in a least squares formulation. In other words, any attempt to
numerically solve Eq. () must be followed by a certain
regularization procedure. In the present study, regularization is performed
by means of truncated SVD that brings about a notion of r-solution .
In brief, the notion of r-solution can be described as follows: any compact
operator A can be described in a Hilbert space with a singular
system {sj, gj, ej}, j = 1, … ∞, where sj ≥ 0
(s1 ≥ s2 ≥ … ≥ sj ≥ …) are singular values and
{gj}, {ej} are the left and the right singular vectors.
Aej = sjgj and sj → 0 with j → ∞. The
systems {gj}, {ej} are orthogonal. A very important
property of the singular vectors is that they form bases in the data and
model spaces, therefore, the solution of Eq. () can be given by
the expression:
φ(x,y)=∑j=1∞η0⋅gjsjej(x,y).
As one can see from Eq. (), the ill-posedness of the
operator equation of the first kind with compact operator is due to the fact
that sj → 0 with j → ∞, i.e. one can perturb the right-hand side
η0(s, t) in such a way that its vanishing perturbation can lead to a
large perturbation of the solution.
The regularization procedure based on truncated SVD leads to a notion of
r-solution given by the formula
φ[r](x,y)=∑j=1rη0⋅gjsjej(x,y).
An r-solution is the projection of the exact solution of Eq. ()
onto a linear span of r right singular vectors
corresponding to the top singular values of the compact operator A.
This truncated series is stable for any fixed parameter r with
respect to perturbations of the right-hand side and the operator as it is
. The value of r is determined by the formula
r=maxk:sk/s1≥1/cond,
where cond = cond(A) is set by the user restriction on the conditioning
number of the matrix A. It is clear that the value of r
depends on the rate of decreasing of singular spectrum of the matrix A,
which is tightly bounded with the parameters of the observation system
and noisiness of data. Indeed, let us assume the perturbation in the
right-hand side η0(s, t) is known
and can be written in the form
ε(t)=∑j=1Lεj(t)gj,
then the perturbed solution will be represented as
φ[r](x,y)=∑j=1rη0+ε⋅gjsjej(x,y).
One should limit the upper index in the last sum from the time when sj is
far less εj(t) to avoid the numerical instability. It is
reasonable that the larger the r is, the more informative the solution will be.
Note that the fitted value of r is much less than a minimum of the matrix dimensions.
The analysis of the singular spectrum of the matrix A is the key aspect
in the proposed methodology because it allows one to predict the recovered
source function φ(x, y) with a certain observation system and bathymetry.
Finite dimensional approximation and r-solution
In a real situation, one can numerically resolve only a finite dimensional
subsystem of Eq. () with (K × L) submatrix. Convergence of the
r-solution of a finite-dimensional system of linear algebraic
equations to the r-solution of an operator equation was carefully
investigated by .
To solve numerically Eqs. ()-() we applied a finite
difference approach for its equivalent first order linear system in terms of
the unknown water elevation η(x,y,t) and velocity vector V:
ηt+g∇⋅(hV)=0Vt+g∇η=0
completed by the following initial conditions:
η|t=0=φ(x,y),V|t=0=0;
and the boundary condition on the solid boundary:
V⋅n=0
and absorbing conditions on the open boundaries. This problem was
approximated by an explicit–implicit four-point finite difference scheme on a
uniform rectangular grid which is based on the staggered grid stencil using
the central-difference approximation of spatial derivatives. As it was
mentioned above, the wave run up is not considered in this study, hence we
infer the coast line when the depth h(x, y) = 50 m.
In order to obtain a system of linear algebraic equations by means of the
projective method, a trigonometric basis was chosen in the model space,
i.e. the unknown function of water surface elevation φ(x, y) is
approximated in the target domain Ω (Sect. 2) by a sum of
spatial harmonics {φmn(x, y) = sin mπl1(x - x1) ⋅ sin nπl2(y - y1)}
with unknown coefficients c = {cmn}:
φ(x,y)=∑m=1M∑n=1Ncmnφmn(x,y),
the center of the tsunami source was believed to be at a point (xc, yc)
being the central point of the domain Ω and l1 = (xM - x1);
l2 = (yN - y1). We assumed the water level oscillations η0(x, y, t)
are known at a set of points {(xp, yp)}, p = 1, …, P for a finite
number of time samples {tj}, j = 1, …, Nt , i.e.
η0=η11,η12,…,η1Nt,η21,…,η2Nt,ηP1,…,ηPNtT,ηpj=η0xp,yp,tj.
The unknown function φ(x, y) was sought for according to Eq. ().
Now, we assume that the dimensions of the solution and the data space are
equal to:
dim(sol)=K=M×N;dim(data)=L=P×Nt.
In the end, a linear algebraic system for the unknown
vector c of coefficients {cmn} (ordered in any
one-dimensional way) from Eq. () was obtained:
η0=Ac,
where A is a matrix whose columns consist of computed waveforms in
every receiver for each spatial harmonic used as the source, η0 is
a vector containing the observed tsunami waveforms. Now, the matrix A
is of K by L matrix. After the SVD procedure one could obtain the
singular values of the matrix A, its left and right singular vectors
{gi, i = 1, …, L}, {ek, k = 1, …, K}. The
value r could be fitted by analyzing the singular spectrum of the matrix A.
Then the r-solution of Eq. () is represented by the sum
c[r]=∑j=1rαjej,
where {αj = (η0,gj)sj} and
{gj}, {ej} are the left and the
right singular vectors of the matrix A, {sj} are its singular values.
Finally, the numerical simulation includes the following steps:
First, we obtain synthetic marigrams in all receivers by solving the
forward problem with a certain function φ(x, y) as a source to be
reconstructed. Thus, the vector η0 in Eq. () is obtained.
The computed marigrams are perturbed by a background noise, i.e. a
high-frequency disturbance and appropriate filtering is applied.
Next, the matrix A is numerically computed by solving the forward
problem with every spatial harmonic {φmn},
m = 1, …, M; n = 1, …, N as a source.
Further, the standard SVD procedure is applied. The analysis of singular
spectrum of the matrix A allows one to choose the number r
varying the conditioning number of the matrix A, as was explained
above, and to compute the coefficients {cmn} by solving Eq. ().
After this, the function φ(x, y) can be computed according to Eq. ().
To estimate the efficiency of the inversion experiment we use a misfit
parameter which is defined as a relative L2-error by the formula:err%=|φ-φ^|L2(Ω)|φ|L2(Ω)×100%.
In the text below, the misfit parameter is denoted by err %.
Numerical experiments: description and discussion
A series of calculations has been carried out by the method proposed to
clarify the dependence of the efficiency of the inversion on certain
characteristics of the observation system such as the number of receivers and
their location, frequency or temporal range of data and the signal-to-noise
ratio. We postulate that source area Ω is given, as it is in real
cases. Synthetic data for the numerical inversion experiments presented below
were computed as a solution of Eqs. ()–() with
respective open boundary conditions. In addition, the function
φ(x, y) was explained by the relation:
φ(x,y)=β(x,y)⋅α(x),
where function β(x, y) makes a semi-ellipsoid with the center at the
point (x0, y0) and the radii Rx and Ry on the plane z = 0. The
parameter α(x) is perturbation of this semi-ellipsoid. When
α(x) = 1 the initial tsunami waveform was assumed to be a semi-ellipsoid
or a semi-sphere.
Case study h(x, y) = const
Firstly, to avoid the influence of such factors as bathymetry and data noise,
we consider a formal calculation domain with open boundaries, with the depth
h(x, y) = h0, the wave phase velocity c(x, y) = gh0 = c0 and
α(x) = 1. The setting of the problem allows us to consider our problem in
the spectral domain and to use the r-solution method for analytical
solution . The basic properties of the inverse operator were
numerically studied.
A layout scheme of the source-receivers arrangement in Model 5.1.
In short, we have shown that the inversion results are better when the
aperture length is rising, increasing the upper limit of the frequency band
makes the obligatory effect for the inversion quality and leads to decreasing
the smearing of the recovered function and, hence, to increasing extreme
values in the center of the source. The results of the inversion by a wide
angle aperture coincide with the experiments on a wide linear aperture. It
was shown, if receivers were uniformly distributed along the linear aperture,
the efficiency of the inversion was improving when the number of receivers
increases up to 15 but further increasing the receivers number was useless.
The receivers-to-source distance has no effect on the inversion result in
the case studied. If the number of receivers is fixed the efficiency of the
inversion rises with the azimuthal coverage. We have shown that a more
precise definition of the target domain relative to the size of a tsunami
source leads to a decrease in the number of necessary receivers, perhaps as
little as 5–7. In addition, the released energy as well as the
maximum value of the recovered function increases. However, the total volume of water
displaced due to the source in the domain Ω is not varied.
The results obtained allowed us to outline the main details of the
methodology proposed. The basic point is that analyzing a singular spectrum
of the obtained matrix A enables us to make an assumption about the
forthcoming efficiency of the inversion.
Let us see how that works in the context of the dependence of the efficiency
of the inversion on the number of receivers and their azimuthal coverage. Our
numerical experiments were conducted with the following parameters (in
spectral domain): Ω = {(x, y): -100 km ≤ x ≤ 100 km;
100 km ≤ y ≤ 200 km}, frequency w ∈ [0.001, 0.01] Hz; α(x) = 1 in
Eq. (), i.e. the type of the source is a semi-ellipsoid with
Rx = 50 km; Ry = 25 km and the center was assumed at the point
(x0, y0) = (0.150) according to Fig. . The conditioning number
of the matrix A was 108 in all calculations that is possible due
noise-free synthetic data. Let us define an above assembly of the fault
parameters as Model 5.1. Our purpose was to obtain acceptable results
of the inversion using a minimum of marigrams, so, we made a computer
simulation when the number of receivers ranged from one to five when they were
placed on the circle aperture with the different aperture angle.
Figure shows a layout scheme of the source-receivers arrangement in Model 5.1.
We will use the following notations: (1) the conditioning number of the
matrix A is designated as cond. (A); (2) the value
100 × max (φ(x, y)), while (x, y) ∈ Ω is denoted by the
symbol {100max}; (3) the misfit parameter is denoted by err %.
The common logarithms of singular values for these numerical experiments are
plotted in Fig. . A sharp decrease in the singular
values, when their number increases, is typical for all calculations in all
cases of the study, due to the ill-posedness of the problem.
Comparing the singular spectra in Fig. we expect the worst
result of inversion when 1–2 receivers were used, while the best result is
provided by using 5 receivers. Moreover, the behavior of singular spectra
makes it possible to expect the improvement of inversion by increasing the
aperture angle for every set of receivers.
The dependence of singular values of the matrix A ( in
logarithmic scale) on their numbers; ρ denotes the number of receivers
used in the inversion and arranged on the aperture with angle equal to α.
As will be clear below, increasing the number of records alone does not lead
to a good inversion if there is insufficient azimuthal coverage with respect
to the source and, on the contrary, in real cases it turns out that the
noisiness of data is raised resulting in lower efficiency of inversion.
Indeed, in Fig. one can see how the number r (the blue
line) and a maximum value of the recovered function (the red line) varied on
the number of receivers used (down horizontal axis) and on the aperture angle (upper
horizontal axis). One can see from these graphs that increasing the number of
receivers, in total, leads to a better inversion: misfit parameter (the green
line) decreases by increasing the number of receivers used, as well as
maximum value of the recovered function tends to a maximum value of the
theoretical function. If the number of receivers is fixed, the efficiency of
inversion rises with the azimuthal coverage that has a good matching with our
previous assumption based on analyzing singular spectra. We have also shown
that if the aperture angle is sufficiently wide, the influence of the
conditioning number is not significant, but the inversion parameters are
worse for a smaller conditioning number of the matrix A. The
recovered functions for the cases discussed above are presented in
Figs. –.
Case study h(x, y) = h(x)
How does a bottom relief and a more realistic type of the source influence
the inversion result? To answer this question we have carried out numerical
experiments for the model bottom topography having some basic morphological
features typical of the island arc regions (see Fig. ).
As the initial sea surface displacement, a sea floor deformation of typical
tsunamigenic earthquakes with reverse dip-slip or low-angle trust mechanisms
was used. Its dipolar shape was simulated according to Eq. ()
with the parameter α(x) = (x - x0 + 3 ⋅ R1) ⋅ (x - x0 + R1/6); and
R1 = 25; R2 = 50 (see Fig. , left panels). The target domain
Ω was a rectangle {(x, y): ∈ [100; 200] × [50; 150]}, the
calculation domain was a rectangle Π = {(x, y): ∈ [0; 300] × [0; 200]},
the center point of the tsunami source was placed at (x0; y0) = (150; 100)
(see Fig. ).
The parameters of the inversion using 1, 2, 3, 5 receivers: 100 max
symbolizes maximum value of the recovered function multiplied by 100 (the red
line), the values of number r (the blue line) and the misfit parameter err %
(the green line).
The recovered functions using 2 (left panels), 3 (middle panels), 5 (right panels)
receivers located on the circle aperture with aperture angle equal to
α = π/5 (left panels, middle panels) and to α = π/10
(right panels), respectively.
The recovered functions using 2 (left panels), 3 (middle panels), 5 (right panels)
receivers located respectively on the circle aperture with aperture angle
equal to α = π.
Synthetic data for the numerical inversion experiments presented below were
computed as a solution of Eqs. ()–(). The full
reflection boundary condition described in Eq. () is fulfilled on
the coast line x = 0 and the absorbing boundary conditions are imposed on the
open free boundaries:
cηyt+ηtt+c22ηxx=0,(x,y)∈y=0;-cηyt+ηtt+c22ηxx=0,(x,y)∈y=200;-cηxt+ηtt+c22ηyy=0,(x,y)∈x=300.
The function φ(x, y) was sought for according to Eq. ()
with M = 25, N = 11. Synthetic marigrams were calculated for Nt = 1990 time
instants. Receivers were disposed on the segment [10; 190] of the line x = 0,
i.e. a maximum of the aperture length was 180.
As in the previous case with a constant depth of the calculated basin we
tried to use a minimum number of marigrams in the inversion process. For this
reason, the parameter P was equal to 1, 2, 3, 5. Some aspects of this
study were presented in . We now turn to these models to
make a generalization. The point in question is the location of receivers
with respect to the dipolar source.
At the initial stage, the numerical experiments were made without noisy data.
Let us name the central point of our linear aperture as a midpoint.
By the midpoint we mean the projection of the central point of the
rectangle Ω (or the central point of the source, which is the same)
onto the aperture line (see Fig. ). The graphs of the misfit
parameter err % when one (the blue line) and two (the red line) receivers
were used in terms of the position of receivers on the aperture line are
plotted in Fig. . The red line in Fig. refers to
the following numerical experiments with two receivers: the first receiver
was fixed at the initial point of the aperture and the second one was moving
to the endpoint of the aperture segment stopping every 10 km along the line
x = 0 with coordinates (0, 10 n) according to the coordinate system of
Fig. . Hence, the length of aperture in every position is equal
to 10 n, n = 1, …, 19.
The misfit parameter err % in terms of the position f receivers
on the aperture: with one receiver used (the blue line); with two receivers
(the red line).
The most important information that should be drawn from these graphs points
to the presence of an obvious minimum of the functions presented. One can see
points of a minimum in these graphs corresponding to the experiments with the
receiver disposed at midpoint. In other words, availability of
waveform from this observation point significantly improves the inversion
result. The following numerical experiments confirm this inference.
A substantial decrease of the misfit parameter when receivers are placed at
the midpoint can be explained by the dipolar shape of the source and,
hence, the signal in this direction is mostly informative.
For this reason, in the following experiments with three and five receivers,
one of receivers is always fixed at the midpoint. The conditioning
number of the matrix A was equal to 100 in the experiments below.
Now, synthetic waveforms were simulated as a result of the solution to the
direct problem described in Eqs. ()–() perturbed by the
background noise, consisting in a high-frequency disturbance about 5 % rate
of a maximum signal amplitude over all the receivers. Admittedly, we did not
obtain any appropriate result with perturbed synthetic data due the ill-posedness of the problem. However, since a tsunami wave is more
lower frequency compared to the background noise, it is reasonable to
apply the frequency filtration of an observation signal. In this paper,
filtration is done by a method of grid function smoothing proposed by
V. A. Tsetsokho and A. S. Belonosov in 1976. The description of this method can be
found in .
The experiments conducted show that a higher rate of the synthetic background
noise results in increasing the misfit parameter with the same conditioning
number of the matrix A in spite of filtration. It is clear that
increasing the conditioning number of the matrix A allows one to use
a larger value r and, therefore, to obtain a more precise solution and a
lesser misfit parameter.
We have considered the following versions of the receivers disposition:
Case1: an observation system of three receivers. One receiver is always
placed at the midpoint but two other receivers are symmetrically placed
with respect to the midpoint with distances 10 n, n = 1, …, 9 every step.
Case2: an observation system of five receivers. Again, one receiver is always
placed at the midpoint, two pairs of receivers moved symmetrically from the
center to the endpoints of the aperture, while a distance between every
nearest-neighbor receivers in every pair was fixed as 40 km (the case named C2.1),
20 km (the case named C2.2) and 10 km (C2.3).
There is an aperture length on the horizontal axis in Fig. .
The parameters of the inversion for Case 1 as functions of the aperture length
are presented in Fig. . The misfit parameter err % (the blue
line) is decreasing but maximum and minimum values of the recovered function
(the orange and the brown lines, respectively) converge to the corresponding
values of the initial function (the red lines) at the moment when the
aperture length approaches the projection of a source on the aperture line
that corresponds to the yellow columns in Fig. .
These regularities remain valid for the inversion with five waveforms.
Indeed, the behavior of the 100max (the magenta line) and err % (the dark
magenta line) for case C2.2 is similar to the ones of case C1 and bears
witness to better results of the inversion.
The inversion parameters for the Case1 (three receivers) and Case2.2
(five receivers). Aperture located symmetrically with respect to midpoint.
The yellow color of columns denote that the aperture is placed inside the
segment of the coast line corresponding to the projection of the source on
the coast.
Case C2.2 differs from case C2.3 by a more uniform distribution of the same
number of receivers within the acting aperture segment being the projection
of the source on the aperture line that leads to improving the inversion
parameters. By comparing the values of parameters in the yellow and
light-blue columns in Fig. one can conclude that a further
increase in the aperture length leads to worse results.
The importance of the receiver location at the midpoint was confirmed
in these series of numerical experiments, too. As was shown, replacing the
central point of a set of receivers from the midpoint results in
increasing the misfit parameter and, at the same time, in decreasing the
maximum value of the recovered function. In other words, we lose the most
informative waveform.
The dependence of singular values of the matrix A (in
logarithmic scale) on their numbers when three (the blue line) and five (the
red line) receivers are used in the inversion (left panel). The recovered function
with (25 × 11) exact coefficients according to Eq. ()
with φmax = 0.73 m, φmin = 0.34 m in target domain
(middle left panel).The recovered function when three waveforms were used in the
inversion: err % = 37.1 %; φ^max = 0.63 m;
φ^min = -0.282 m; length of aperture. = 80 km; (middle right panel). The recovered function
when five waveforms were used: err % = 20.4 %; φ^max = 0.71 m;
φ^min = -0.289 m; length of aperture. = 140 km (right panel).
As an example, the recovered functions corresponding to the inversion for
Case 1 and Case 2.2 are presented in Fig. (middle-right and
right panels). From the graphs of singular spectra of the inversions with three and
five waveforms plotted in Fig. (left panel) one can expect that the
inversion in the latter case will be more successful. Indeed, results of
numerical experiments presented in Fig. (middle, right panels)
substantiate our assumption based on analyzing singular spectra.
It is now clear that the quality of inversion strongly depends on the
disposition, the number of receivers and noisy data. The robust result could
be obtained when, at least, one of the receivers involved is placed at the midpoint which is a projection of the major variability direction of the source.
Case study: real bathymetry of the Peru subduction zone
Finally, to illustrate some of the ideas let us consider the results obtained
for the case study with the real bathymetry of the Peru subduction zone. We
are interested in how distinctive features of a real bathymetry affect the
inversion process. The simulation area is located from 85 to
71∘ W and from 5 to 15∘ S. We set up the
following parameters for our calculations: the domain Φ was the aquatic
part of a rectangle Π = {0 ≤ x ≤ 600; 0 ≤ y ≤ 400} with
piecewise-linear boundaries of the dry land, the domain {Ω = {400 ≤ x ≤ 500;
200 ≤ y ≤ 300} (see Fig. ). As mentioned
above, the wave run up was not considered. The observed data were simulated
as a solution of Eqs. ()–() and completed with the full
absorbing boundary conditions of second order of accuracy being fulfilled on
the open boundaries.
Again, as an initial sea surface elevation, the sea floor deformation of
typical tsunamigenic earthquakes with reverse dip-slip or low-angle thrust
mechanisms was used and it is plotted in Fig. (left panel) with the
following parameters: the center point (x0; y0) = (450; 250), maximum and
minimum values of initial displacement in source area φ(x, y)φmax = 1.959 m;
φmin = -0.67 m. As before, we tried to
obtain acceptable results of the inversion using a minimum number of records.
In our calculations, the function φ(x, y) was sought for according to
Eq. () with M = 25, N = 11. These values are defined by the
shape of theoretical function φ(x, y). Depth h(x, y) was assumed to
be the real bathymetry of the Peru subduction zone. We placed 14 receivers at
points in the domain Φ which are enumerated according to
Fig. and are marked by the green color (∘), i.e. P = 14
in Eq. (). The time interval was long enough for the tsunami
wave to reach all receivers, specifically, the number of time steps was
Nt = 1684 in the case presented.
Isolines of the Peru subduction zone with depth values (in km), the
target domain and 14 receivers marked by the green color symbols (∘).
After specifying all the parameters we carried out the steps of
the numerical simulation mentioned in Sect. 4. Synthetic marigrams were
perturbed by the background noise with the disturbance rate about 3 % of a
signal maximum amplitude over all the receivers. It is imperative that the
filtration procedure was made after the noise pollution. Next, the standard
SVD-procedure was performed and the singular spectrum of the
matrix A was obtained.
As before, first of all we analyzed the singular spectrum of the matrix A.
For example, the graphs of the common logarithm of the singular values
of the matrix A corresponding to their numbers are presented in
Fig. . for the inversion with three (the red line), four (the
blue line), nine (the magenta line) and ten (the green line) waveforms used
in the calculations. Comparing singular spectra for the inversion with some
sets of three and four marigrams in Fig. , one can assume that
such an increase in the number of receivers will lead to the deterioration of
the solution. Inversion parameters corresponding to these cases are presented
in Table (the first and the fifth rows).
We have seen, until now, that adding new receivers always led to better
solutions when a depth of the calculation basin was a synthetic function.
However, this prediction is not universally true for the case with a real
bathymetry. Only increasing the number of receivers without considering their
azimuthal coverage could result in a worse inversion due to rising a general
level of noise pollution of data.
Typical graphs (upper parts) of singular values in the common
logarithm scale of the matrix A with respect to their numbers when
the number of waveforms used in the inversion is equal to three (the red
line), four (the blue line), nine (the magenta line) and ten (the green
line).
The inversion parameters with sets of used receivers.
P
Cond
r
Err %
φmax
φmin
Receivers
3
100
41
71.7
1.213
-0.78
3, 10, 12
3
1000
42
74.8
1.577
-1.55
3, 10, 12
4
100
15
74.22
0.892
-0.655
3, 4, 12, 13
4
1000
16
75.22
0.905
-0.647
3, 4, 12, 13
4
100
34
99.1
0.128
-0.107
3, 4, 5, 7
4
100
34
65.86
0.419
-0.156
2, 6, 10, 14
5
100
45
62.3
1.185
-0.67
3, 4, 6, 7, 11
5
1000
57
40.9
1.699
-0.75
3, 4, 6, 7, 11
5
1000
16
74
0.845
-0.69
3, 4, 6, 7, 9
9
100
73
37.7
1.539
-0.72
3, 4, 5, 6, 7, 8, 9, 10, 11
9
1000
104
63.3
2.396
-1.29
3, 4, 5, 6, 7, 8, 9, 10, 11
9
1000
104
59.9
2.25
-1.04
3, 4, 5, 9, 10, 11, 12, 13, 14
10
100
72
37.24
1.545
-0.645
3, 4, 5, 6, 7, 8, 9, 10, 11, 12
10
1000
99
86.75
2.632
-1.909
3, 4, 5, 6, 7, 8, 9, 10, 11, 12
10
1000
15
62.9
1.099
-0.74
1, 2, 3, 5, 6, 7, 9, 12, 13, 14
The number of receivers used in the inversion process is denoted
by P. Maximum and minimum values of the recovered function are symbolized by
φ^max and φ^min respectively (scaled in
meters) The dimension of the subspace where the exact solution was projected
is symbolized by r. The conditioning number of the matrix A
is denoted by cond and the misfit parameter is denoted by err %.
Obviously, the parameter r should be taken only from the first
interval, where the common logarithms of singular values are slightly
sloping because further increases in number r leads to the solution
instability. On the other hand, r should be large enough to provide a
suitable spatial approximation of φ(x, y). From the numerical
experiments it is clear that a satisfactory parameter value has
r ≥ 70 .
Some results of our numerical simulations confirming these inferences are
presented in Table . Analyzing the values of the inversion
parameters in Table , one can see that the inversion with five
marigrams (the eighth row) is better than the inversion with nine or ten
marigrams (the 11th, 12th, 14th and 15th rows). To compare the inversion
parameters in Tables and one has to look at
Fig. . It becomes clear that the key role in improving the
quality of the inversion is played by locating the monitoring system relative
to the topography and source area.
Singular spectra in the common logarithm scale for the models: V1.1
and V1.2 (the brown line), V2.1 and V2.2 (the red line); V3.1 and V3.2 (the
blue line), V4 (the green line) (see Table 2).
Indeed, adding to the observation system the receivers with numbers {12,
13, 14} (see the 12th and the 15th row in Table ) which were
not affected by disturbance due to the reflection from the underwater
vertical ledge, did not improve the solution, according to the expectations.
The same is true for a set of receivers {1, 2, 3, 4} which, on the
contrary, were placed in the direction where the distortion of the signal is
more possible due to the features of the relief. Using receivers with numbers
{5, 6, 7, 8, 9, 10, 11} one can obtain a strong improvement of the
inversion. The replacement of even one or two of them leads to a solution
deterioration (see the eighth and the ninth rows in Table ).
What are the odds? It seems reasonable to say that the matter is in the dipole
shape of our source and in the existence of a certain angle between its axis
and the line of the reflection (the coast line and the underwater ledge). The
axis of the source in our case is directed along the axis x and is not
perpendicular to the coast line like in the previous model. The latter set of
receivers is distributed along the reflection ray corresponding to the
direction of the strongest variability of the source. It should be remarked
that the use of all the receivers does not improve the inversion because this
set of marigrams involves many fewer informative waveforms that only leads to
increasing the noisy pollution of the data.
The last series of the numerical experiments was aimed at finding out the
most informative direction of the receivers location in terms of improving
the inversion. We have chosen several sets each consisting of seven receivers
differing in their location.
The inversion parameters for different sets with seven receivers.
Model
Cond
r
Err %
φmax
φmin
Receivers
V1.1
100
23
62.8
1.106
-0.696
3, 4, 5, 6, 7, 8, 9
V1.2
10 000
32
51.9
1.392
-0.49
3, 4, 5, 6, 7, 8, 9
V2.1
100
73
37.7
1.559
-0.717
3, 4, 5, 6, 9, 10, 11
V2.2
10 000
92
32.7
1.686
-0.735
3, 4, 5, 6, 9, 10, 11
V3.1
100
65
45.9
1.449
-0.923
5, 6, 7, 8, 9, 10, 11
V3.2
10 000
104
26.7
1.816
-0.703
5, 6, 7, 8, 9, 10, 11
V4
10 000
59
31.73
1.706
-0.737
1, 3, 5, 7, 9, 11, 13
Maximum and minimum values of the recovered function are denoted
by φ^max and φ^min respectively (scaled in
meters). The dimension of the subspace where the exact solution was projected
is symbolized by r. The conditioning number of the matrix A
is denoted by cond and the misfit parameter is denoted by err %.
The recovered functions: cond(A) = 100 (top panels) by models V1.1
(left panel), V2.1 (middle panel) and V3.1 (right panel); cond(A) = 10000 (bottom panels) by
models V1.2 (left panel), V2.2 (middle panel) and V3.2 (right panel).
The upper parts of singular spectra for these numerical experiments are
plotted in common logarithm scale in Fig. . It is also shown
in Fig. , how one could define the number r for every
settled conditioning number of the matrix A.
In Table one can see the major parameters of the inversion with
seven marigrams by Models V1–V4 which differ in the receivers
location and the conditioning number of the matrix A.
The recovered functions for models V1.1, V2.1, V3.1, V1.2, V2.2, V3.2 are
plotted in Fig. . It is clear, the use of the large
conditioning number of the matrix A and, as a consequence, a large
value of r in the liable interval leads to a lesser smearing effect
in the shape of the recovered function.
Theoretical source model: φmax = 1.959 m;
φmin = -0.67 m (left panel). Model V3.2 φ^max = 1.816 (1.514) m;
φ^min = -0.707 (-0.548) m; r = 104; cond(A) = 104
(right panel). The values in parentheses are extreme values of the recovered
waveform after smoothing.
Comparison between theoretical (the red line) and calculated
waveforms in all receivers: the green line indicates the waveform computed
using the source reconstructed with 5 receivers {3, 4, 6, 7, 10}; the
blue line indicates the waveform computed using the source reconstructed with
7 receivers {5, 6, 7, 8, 9, 10, 11}; the time increment equals 3 s.
It should be noted that the misfit parameter gives only a global estimation
of the efficiency of the inversion. Based on our experiments with perturbed
data and a real bathymetry we can conclude that the value of the misfit
parameter err % ≈ 20–26 % allows us to obtain a robust shape of the
recovered source. In Fig. the theoretical function
φ(x, y) and the one recovered by model V3.2 are presented. The extreme values
of the recovered function have changed due to smoothing.
After the inversion by model V3.2 was completed, we again solved the direct
problem with the recovered and smoothed function φ^(x, y) and
calculated marigrams at the same 14 points. Marigrams computed with the
recovered tsunami source have a good matching with the initial synthetic ones
not only in seven receivers used in the inversion process but also in all
14 receivers as well (see Fig. ).
This fact can be treated as a verification of the inversion algorithm. In
addition, the coincidence of marigrams has a great importance for predicting
water elevation in the area relying on data provided by some monitoring system.